Several approaches have been adopted in simulation of ecological and biological systems. Usually the initial approach to simulation of an ecosystem is to develop a simple linear model. That is to assume that if Y1=f(x1) and Y2=f(x2) then (Y1+Y2)=f(x1+x2) or, that no second interaction between the two variables controlling a process. Often this is not true, and linear model may only give an approximation (reasonable or otherwise) to the real system. Inherent in biological systems are non-linearities. Simulation models may be stochastic or deterministic. This simply denotes whether a model contains some random events or not. Deterministic models contain a sequence of events, which follow with complete certainty to produce a result not affected by chance events. Conversely, if one or more of the sequence pathways are selected on a probability basis, the model is stochastic or probabilistic. The mathematical approach to a computer model defines a number of system variables, which characterize and control the xe2x80x98flowxe2x80x99 between component segments of the model. The flow or interaction between component segments of the model is derived from transfer functions. The model driving equations (forcing functions) involve those external parameters required as input to the model but which are little affected by the model if at all.
Two basic strategies are used for modeling. Compartment models, which are usually simple linear systems of differential equations very often describing biomass changes or energy flow in a system. Their purpose is to analyze the performance of the model rather than necessarily to attempt to model the real system. The second and more detailed strategy that has been employed is the component approach of Holling (1969) (detailed in an article titled xe2x80x9cThe strategy of building models of complex ecological systemsxe2x80x9d in a book titled xe2x80x9cSystems analysis in ecologyxe2x80x9d by Watt K E F, (ed.) published by Academic Press N.Y. pages 195-214); or the building block by Kershaw and Harris (1969) (detailed in an article titled xe2x80x9cSimulation studies and ecology: A simple defined system modelxe2x80x9d in Statistical Ecology volume 3: pages 1-21, published by Penn. State Univ. Press). This follows a stepwise fashion, from an experimental examination of each block or component, in turn, to a series of equations relating the parameters involved in each component, and back to the experimental-testing of the model at each step.
High biocomplexity, high natural dynamics, and certain periodic processes such as habitat fragmentation characterize ecological and biological systems. While empirical/statistical models describe the global behavior of ecological and biological systems and models of differential equations try to represent single processes, there is another type of knowledge that handles processes and behavior patterns in a causal manner. This knowledge cannot be formalized in generic predicate logic or similar paradigms without losses. Therefore a new approach is required that will match the changing dynamics of ecosystems with high adaptability and built-in feed back. The processing engine of the foregoing may be implemented through artificial neural networks comprising a plurality of logic elements called neural circuits. A neuron is the fundamental building block of an artificial neural network. The computer model represents these neurons as well as the whole network by data structures. The data are structured in layers. Each layer or data set represents one or more neurons. The neurons are connected with each other and with the surroundings. The neuron has multiple inputs and a single output.
There are many types of neural network architectures. Such neural architectures as xe2x80x9cback propagationxe2x80x9d, xe2x80x9cperceptronxe2x80x9d and xe2x80x9cHopfield networkxe2x80x9d are the best known. Other neural network structures have been discussed extensively in a book titled xe2x80x9cSimulating Neural; Networksxe2x80x9d published in 1994 by Verlag Vieweg and authored by Norbert Hoffmann. The structure comprises of three or more layers, neurons connected to the input set and form a layer of input neurons, others transfer their output to the output set, and are called output neurons. The remaining neurons are not connected to the surroundings, and are called the hidden neurons. Each neuron in the hidden layer multiplies its inputs, as received from the input nodes, by a given weight to produce a product.
Most neural network structures have serious drawbacks. Which include time-consuming training of the networks for relatively complex problems such as that for ecological and biological systems simulation.
Another, disadvantage is that when weights converge, they usually converge to local minima, which gives erroneous solution. For example, a particular function may become slightly larger, regardless of how a particular parameter is moved. However, if the parameter were to be moved into a completely different place, the loss function may actually become smaller.
One can think of such local minima as local xe2x80x9cvalleyxe2x80x9d or minor xe2x80x9cdentsxe2x80x9d in the loss function. However, in most practical applications, local minima will produce xe2x80x9coutrageousxe2x80x9d and extremely large or small parameter estimates with very large standard errors. In such cases different start values have to be specified and tried again.
To avoid local minima, statistical methods such as Boltzman training or Cauchy training has been applied. However, the optimum solution is the xe2x80x9cbest fitxe2x80x9d global minimumxe2x80x9d for a given set of examples. The U.S. Pat. No. 5,781,701 to Wang 1998, establishes a method that uses a neural network which utilizes a plurality of neuron circuits which do not individually utilize any non-linear function or summing circuit and which each require only a multiplier circuit as its main processing element.
In summary, the present approach of simulating ecological and biological systems using conventional methods do not match the high biocomplexity, high natural dynamics, and periodicity that characterize such systems. Conventional neural network approach needs to be modified to fit the inherent natural biological and physiological processes between elements of the ecosystem model. In addition, the complex circuitry of conventional neural networks severely limits their implementation in the form of computer software, and hence its application in ecological system modeling.
What is therefore required is a straight-forward neural architecture that is easy to implement in form of a software which yields a global minimum to each given set of input vectors and does not require repetitive training.
A preferred embodiment of a neural network designed in accordance with the teachings of the present invention comprising input neurons and three layers of neurons by way of example. Input neurons serves an interconnect function, connecting external inputs to the network. A first layer of hidden neurons comprising neurons not connected to the surroundings. The second layer of hidden neurons receives inputs from the first. A third layer of neurons transfer their output values to the output set. They are called the output neurons.
In contrast to conventional artificial neural networks, a neural network constructed for ecological system scenarios in accordance with the present invention converges on a global solution using standard statistical regression model estimation, which can often be computed in a few minutes on a personal computer.
Moreover, in contrast to conventional approaches, there is provided in accordance to the teachings of the present invention a neural network which utilizes a plurality of neurons. The first layer of hidden neurons use non-linear estimation to predict the neuron weights from driving independent variables. The weights have established biological relationship with the neuron output.
Thus it will be appreciated that a neural network constructed in accordance with the present invention performs with accuracy, in less computational time and reduced cost and complexity of implementation, whether in a computer program or hardware design.
In addition, a neural network for ecological systems constructed in accordance with the present invention can have single or multiple outputs by providing multiple summing circuit for summing the outputs of the neurons.
Thus it is an advantage of the present invention to provide a neural network for ecosystems which utilizes a plurality of neurons, so that a neural network may be built comprising a very large number of such neurons processing inputs from a plurality of driving variables, resulting in a model which can simulate the high complexity and high temporal dynamics inherent in ecological and biological systems.
It is also an advantage of the present invention to provide a neural network for ecosystem modeling, which does not require repetitive training.
Yet another advantage of the present invention is to provide a neural network for ecosystem modeling which yields a global minimum to each given set of input variables.
It is also another advantage of the present invention to provide a method of modeling ecological and biological systems using a neural network in accordance with the present invention.
According to one aspect of the invention, there is provided a neural network for ecosystem modeling having a plurality of network inputs and at least one network output, the neural network comprising: a plurality of neurons, each neuron having a plurality of inputs and generating an output.
According to another aspect of the present invention, there is provided a method for training neural network comprising a plurality of neurons, which method requires estimation of loss function (to find the best fitting set of parameters) and to estimate the standard errors of parameter estimates via using algorithms (e.g quasi-Newton, Simplex, Hooke-Jeeves pattern moves, and Rosenbrock pattern search) to solve the values of each neuron weight and and hence output value.
According to yet another aspect of this invention there is provided a neural network for ecosystem modeling comprising: a plurality of network and at least one output; a plurality of neurons, each neuron receiving a plurality of inputs and generating an output; a method of operating the neural network, the method comprising the following steps: an initial iterative procedure comparing driving input variables (independent variables) to the weights (dependent variable), at each step, the program evaluates whether the fit of the model to data has improved from the previous step, i.e., how much xe2x80x9cerrorxe2x80x9d was lost between the previous and the current iteration, and calculating the loss function to determine how the goodness of the model fit to the data.
According to yet another aspect of the invention there is provided in a neural network for ecosystem modeling comprising: a plurality of network inputs and at least one network output; a plurality of neurons, each neuron receiving a plurality of inputs applied to the network, reproduces the network using a current model, and compares the output values with given target values and xe2x80x9chierarchially relatesxe2x80x9d (means that the current model is identical to the previous model with the exception of an addition or deletion of one or more driving or independent variables) to the previous model and using the comparison between the goodness of fit for the two models or difference to set the learning rules.